The normalization factor conventionally used in satellite geodesy appears in
[Kaula's Eq. 1.34]{https://store.doverpublications.com/0486414655.html}.
Here I call it $N_{n,m}$:
\begin{align*}
N_{n,m} = \sqrt {(2-\delta_{0,m})(2n+1) \dfrac {(n-m)!} {(n+m)!}}
= \begin{cases}
\sqrt {(2n+1) \dfrac {(n-m)!} {(n+m)!}}, & \quad m = 0
\\
\sqrt {2 (2n+1) \dfrac {(n-m)!} {(n+m)!}}, & \quad m > 0
\end{cases}
\end{align*}
TL;DR
Why normalize?
One compact formula for gravitational potential is
\begin{align*}
U
& =
\frac {G M}{r}
\left[
1 + \sum_{n=1}^{\infty}
\left(\frac {a_e} {r}\right)^n
\sum_{m=0}^{n}
P_{n,m}(\cos\theta)
\big[
C_{n,m} \cos(m \lambda) + S_{n,m} \sin (m \lambda)
\big]
\right]
,
\end{align*}
where coordinates $(r,\theta,\lambda)$ are respectively the
radial distance, colatitude, and longitude
of the point where potential is to be evaluated,
all expressed in a coordinate system fixed in
the body and rotating with it.
Product $G M$ is the body's gravitational parameter.
Coefficients $C_{n,m}$, $S_{n,m}$ encode the body's gravitational field.
The $P_{n,m}(\cos\theta)$ are Associated Legendre Functions (ALFs).
Their argument $\cos\theta$ is the cosine of colatitude $\theta$
measured southward from the north pole,
or equivalently the sine of latitude measured
north and south from the equator.
Degree $n$ and order $m$ are non-negative integers
$0, 1, 2, \dots$ with $0 \le m \le n.$
Reference distance $a_e$
and body mass $M$
make $C_{n,m}$, $S_{n,m}$ dimensionless pure numbers.
For large $n$ or $m$,
ALFs can become very large
while coefficients $C_{n,m}$, $S_{n,m}$ become very small.
To compensate,
a normalization factor (denoted above as $N_{n,m}$) is introduced
which becomes small with large $n,m$.
The factor multiplies $P_{n,m}(\cos\theta)$
and divides $C_{n,m}$ and $S_{n,m}$
so that products $C_{n,m} P_{n,m}(\cos \theta)$
and $S_{n,m} P_{n,m}(\cos \theta)$ are unchanged
within the potential formula.
An overline above $P$, $C$, and $S$ distinguishes
"fully normalized" quantities
from their unnormalized counterparts:
\begin{align*}
C_{n,m} \, P_{n,m}(\cos\theta)
& = (C_{n,m} / N_{n,m}) (N_{n,m} P_{n,m}(\cos\theta))
\equiv \overline{C}_{n,m} \, \overline{P}_{n,m}(\cos\theta)
,
\\
S_{n,m} \, P_{n,m}(\cos\theta)
& = (S_{n,m} / N_{n,m}) (N_{n,m} P_{n,m}(\cos\theta))
\equiv \overline{S}_{n,m} \, \overline{P}_{n,m}(\cos\theta)
.
\end{align*}
The only change to the potential formula is the overlines:
\begin{align*}
U
& =
\frac {G M}{r}
\left[
1 + \sum_{n=1}^{\infty}
\left(\frac {a_e} {r}\right)^n
\sum_{m=0}^{n}
\overline{P}_{n,m}(\cos\theta)
\big[
\overline{C}_{n,m} \cos(m \lambda) + \overline{S}_{n,m} \sin (m \lambda)
\big]
\right]
.
\end{align*}
